CORRIAS, Lucilla , PERTHAME, Benoit, ZAAG, Hatem Global Solutions of some Chemotaxis and Angiogenesis Systems in high space dimensions
Référence: DMA-03-15
Résumé, Abstract : We consider two simple conservative systems of parabolic-elliptic and parabolic-hyperbolic type arising in modeling chemotaxis and angiogenesis. Both systems share the same property that when the $L^{\frac d2}$ norm of initial data is small enough, where $d\ge 2$ is the space dimension, then there is a global (in time) weak solution that stays in all the $L^p$ spaces with $\max\{1 ; \frac d2 -1\}\le p<\infty$. This result is already known for the parabolic-elliptic system of chemotaxis, moreover blow-up can occur in finite time for large initial data and Dirac concentrations can occur. For the parabolic-hyperbolic system of angiogenesis in two dimensions, we also prove that weak solutions (which are equi-integrable in $L^1$) exist even for large initial data. But break-down of regularity or propagation of smoothness is an open problem.
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